If $\frac{32x^2+186x}{(x^2+1)(x+5)}=\frac{37x+1}{x^2+1}+\frac{\lambda}{x+5}$,then $\frac{\lambda}{2}$ is equal to

For $|x| < \frac{1}{2}$,if the coefficient of $x^{10}$ and the constant term in the expansion of $\frac{2 x^3+8 x^2-2 x-2}{(1-x)(1+x)(1-2 x)}$ in powers of $x$ are $l$ and $m$ respectively,then $lm=$

What is the coefficient of $x^4$ in the expansion of $\frac{3x}{(x - 2)(x + 1)}$?

If the partial fraction decomposition of $\frac{x^2+1}{x^3+3x^2+3x+2}$ is $\frac{A}{x+2} + \frac{Bx+C}{x^2+x+1}$,then find the value of $A-B+C$. Note: The original expression provided in the prompt was corrected to the standard form $\frac{A}{x+2} + \frac{Bx+C}{x^2+x+1}$.

If $\frac{x^2}{(x^2 + a^2)(x^2 + b^2)} = k \left( \frac{a^2}{x^2 + a^2} - \frac{b^2}{x^2 + b^2} \right)$,then $k =$

If $\frac{1}{x(x^2 + 1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1}$,then $(A, B, C) = $